Lecture 12
Physics II
Chapter 31
Kirchhoff’s Laws
Course website:
http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsII
Lecture Capture:
http://echo360.uml.edu/danylov201415/physics2spring.html
While still only a graduate student, he published a paper that included a pair of rules for the analysis of circuits (Kirchhoff’s laws of circuits).
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Kirchhoff’s Law
Some circuits are too complicated to analyze
(none of the elements are in series/parallel)
Kirchhoff’s rules are very helpful.
To analyze a circuit means to find:
1. ΔV across each component
2. The current in each component
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Kirchhoff’s Junction Law
For a junction, the law of conservation of current requires that:
At any junction point, the sum of all currents entering the junction
must equal the sum of all currents leaving the junction.
3
out
2
1
in
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Kirchhoff’s Loop Law
For any path that starts and ends at the same point:
The sum of all the potential differences encountered
while moving around a loop or closed path is zero.
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ΔV across a battery
according to a property of a battery
Lower V
Higher V
Δ
Travel direction
Final point For a battery, the potential difference is positive if your chosen loop Initial point
according to a travel direction
direction is from the negative terminal toward the positive terminal
Higher V
Lower V
Δ
Travel direction
Initial point
Final point
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The potential difference is negative if the loop direction is from the positive terminal toward the negative terminal
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ΔV’s across resistors
Lower V
Higher V
(Because I flows from higher V to lower V)
Current direction
+
_
Travel direction
Final point
Initial point
according to a travel direction
Δ
For a resistor, apply Ohm’s law; the potential difference is negative (a decrease) if your chosen loop direction is the same as the chosen current direction through that resistor
Current direction
_
+
Travel direction
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Δ
For a resistor, apply Ohm’s law; the potential difference is positive (an increase) if your chosen loop direction is opposite to the chosen current direction through that resistor
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Example 31.1. Analyze the circuit
No junction points
Loop rule
=
+
(If our assumption turns out to be wrong, the current will be negative)
2) Choose a travel direction (say, CW) and a start point
Now we can find pot. differences across each resistor
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‐
Travel direction
1) Assume CW direction of current
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=
‐=
+
ConcepTest 1
Loop rule
What is ΔV across the
A) 0V
unspecified circuit element?
B) 1V
C) 2V
D) 3V
+12 V +ΔV - 8 V - 6 V = 0
ΔV= 2 V
Travel direction
Multi-Loop
Circuit
Let’s take a look at how the junction rule and loop rule
help us solve for the unknown values in multi-loop circuits.
In general: if there are N junctions
in a circuit, then there are N-1
independent junction equations
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Loop rule
I
Travel direction
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I
Travel direction
Tactics: Using Kirchhoff’s Rules
1. Label the current in each separate branch of the given circuit with a different subscript, such
as I1 , I 2 , I 3 Each current refers to a segment between two junctions. Choose the direction of
each current, using an arrow. The direction can be chosen arbitrarily: if the current is actually in
the opposite direction, it will come out with a minus sign in the solution.
2. Identify the unknowns. You will
unknowns. You may write down more
equations will be redundant (that is,
information). You may use for each
unknown
need as many independent equations as there are
equations than this, but you will find that some of the
not be independent in the sense of providing new
resistor, which sometimes will reduce the number of
3. Apply Kirchhoff’s junction rule at one or more junctions.
3. Apply Kirchhoff’s loop rule for one or more loops: follow each loop in one direction only.
Pay careful attention to subscripts, and to signs:
(a) For a resistor, apply Ohm’s law; the potential difference is negative (a decrease) if your
chosen loop direction is the same as the chosen current direction through that resistor; the
potential difference is positive (an increase) if your chosen loop direction is opposite to the
chosen current direction.
(b) For a battery, the potential difference is positive if your chosen loop direction is from the
negative terminal toward the positive terminal; the potential difference is negative if the loop
direction is from the positive terminal toward the negative terminal.
4.Solve the equations algebraically for the unknowns.
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Thank you
See you on Friday
95.144 Danylov Lecture 12
Department of Physics and Applied Physics